Lemoine circles | second Lemoine circle

Draw antiparallels through the symmedian point K. The points where these lines intersect the sides then lie on a circle, known as the cosine circle (or sometimes the second Lemoine circle). The chords Q_2 P_3, Q_3 P_1, and Q_1 P_2 are proportional to the cosines of the angles of Δ A_1 A_2 A_3, giving the circle its name. In fact, there are infinitely many circles that cut the side line chords in the same proportions. The centers of these circles lie on the Stammler hyperbola (Ehrmann and van Lamoen 2002). The cosine circle is a special case of a Tucker circle with λ = 0.

Brocard circle | Brocard points | excosine circle | Miquel point | second Brocard circle | Taylor circle | Tucker circles